The ideal gas equation

A tyre on a hot day

Pump a bicycle tyre in the morning and ride in the hot sun. By afternoon the tyre feels harder. Nobody added air. The air got warmer, and its pressure went up.

Squeeze the end of a bicycle pump with your thumb over the hole. The air gets harder to push. Its volume went down, and its pressure went up.

A gas changes its pressure, volume and temperature together. Let us find the rule.

Four simple laws

Experiments on gases at low density give four laws.

• Boyle's law. For a fixed amount of gas at a steady temperature, $pV$ stays the same. Halve the volume and the pressure doubles.
• Charles's law. For a fixed amount at a steady pressure, $V \propto T$, with $T$ in kelvin.
• The pressure law. For a fixed amount in a fixed volume, $p \propto T$.
• Avogadro's law. At the same pressure and temperature, $V \propto n$, the number of moles.

The ideal gas equation

All four fit into one equation:

Here $n$ is the number of moles and $R = 8.31$ J/(mol K) is the universal gas constant. A gas that follows this equation exactly is an ideal gas. Real gases come close at low pressure and high temperature.

We can count molecules instead of moles. The number of molecules is $N = nN_A$. Then

For a fixed amount of gas, $nR$ does not change, so

Two rules for every gas problem.

1. The temperature is always in kelvin: $T = t + 273$, with $t$ in degrees Celsius.
2. The pressure is the absolute pressure. A tyre gauge shows the pressure above the air outside. Add the outside pressure, about $101$ kPa, to get the absolute pressure.

The gas on a pressure-volume graph

Keep the temperature fixed and plot $p$ against $V$. Boyle's law gives a curve called an isotherm. A hotter gas gives an isotherm that lies higher and further out.

Move the volume slider. The point slides along the curve. Now change the temperature. The whole curve moves. Does the last readout, $pV/(nT)$, ever change? It stays at $8.31$ J/(mol K). That number is $R$.

Mass, moles and density

The number of moles is the mass divided by the molar mass: $n = m/M$. Put this into $pV = nRT$ and use $\rho = m/V$:

At the same pressure and temperature, a gas with a larger molar mass is denser.

A worked example

Why does a helium balloon rise? Find the density of helium and of air at $300$ K and $101.3$ kPa. Helium has $M = 0.004$ kg/mol and air has $M = 0.029$ kg/mol.

1. We want the density, so $\rho = \dfrac{pM}{RT}$. Start with $\dfrac{p}{RT} = \dfrac{101.3 \times 10^3\ \text{Pa}}{8.31\ \text{J/(mol K)} \times 300\ \text{K}} = 40.6\ \text{mol/m}^3$. That is the number of moles in each cubic metre, for any gas.
2. Helium. $\rho = 40.6\ \text{mol/m}^3 \times 0.004\ \text{kg/mol} = 0.163\ \text{kg/m}^3$.
3. Air. $\rho = 40.6\ \text{mol/m}^3 \times 0.029\ \text{kg/mol} = 1.18\ \text{kg/m}^3$.
4. Check. Air is about seven times denser, the same as $29/4$. So the air pushes the balloon up more than the helium weighs.

Practise with new numbers

Try it in Python

A weather balloon is filled at the ground and let go. As it rises, the air outside gets thinner and colder. The program uses $\dfrac{p_1V_1}{T_1} = \dfrac{p_2V_2}{T_2}$ at each height.

Common mistakes

• Using degrees Celsius. At $0\ ^\circ\text{C}$ the gas would have no volume. Always change to kelvin.
• Using the gauge pressure. A tyre at a gauge pressure of $200$ kPa holds air at about $301$ kPa absolute.
• Mixing litres and cubic metres. In $pV = nRT$ with $R = 8.31$ J/(mol K), use pascal and cubic metres. $1$ L is $10^{-3}$ m³.
• Forgetting that two things change. When a gas rises in the air, both $p$ and $T$ change. Use $p_1V_1/T_1 = p_2V_2/T_2$.

Remember these

• Boyle: $pV$ constant. Charles: $V \propto T$. Pressure law: $p \propto T$. Avogadro: $V \propto n$.
• $pV = nRT = Nk_BT$. $R = 8.31$ J/(mol K). $k_B = 1.38 \times 10^{-23}$ J/K.
• Fixed amount: $p_1V_1/T_1 = p_2V_2/T_2$.
• Density of a gas: $p = \rho RT/M$.
• Kelvin and absolute pressure, always.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A gas in a syringe is squeezed slowly to one third of its volume, at a steady temperature. What happens to its pressure?

2. A sealed jar of air at $7\ ^\circ\text{C}$ is warmed to $77\ ^\circ\text{C}$. By what factor does its pressure rise?

3. What volume does $0.50$ mol of a gas fill at $400$ K and $83.1$ kPa?

4. At the same pressure and temperature, which is denser: nitrogen ($M = 28$ g/mol) or carbon dioxide ($M = 44$ g/mol)? By how many times?